Conventions

1.1 Conventions

I will use a 4-metric $gμν$ with signature $(− ,+, +, +)$ . Following the MTW [80

] conventions, I define the Riemann tensor as

The Ricci tensor is defined as

and the Einstein tensor as

A spatial 3-metric will be written as $γij$ and the Riemann and Ricci tensors associated with it will be defined by (1

) and (2

). Four-dimensional indices will be denoted by Greek letters, while 3-dimensional indices will be denoted by Latin letters.

To avoid confusion, the covariant derivative and Ricci tensor associated with $γ ij$ will be written with over-bars – $¯ ∇j$ and $¯ Rij$ . I will also frequently deal with an auxiliary 3-dimensional space with a metric that is conformally related to the metric $γij$ of the physical space. The metric for this space will be denoted $&tidle;γij$ . The covariant derivative and Ricci tensor associated with this metric will be written with tildes – $∇&tidle;j$ and $R&tidle;ij$ . Other quantities that have a conformal relationship to quantities in the physical space will also be written with a tilde over them.